Metamath Proof Explorer


Theorem ltleni

Description: 'Less than' expressed in terms of 'less than or equal to'. (Contributed by NM, 27-Oct-1999)

Ref Expression
Hypotheses lt.1
|- A e. RR
lt.2
|- B e. RR
Assertion ltleni
|- ( A < B <-> ( A <_ B /\ B =/= A ) )

Proof

Step Hyp Ref Expression
1 lt.1
 |-  A e. RR
2 lt.2
 |-  B e. RR
3 ltlen
 |-  ( ( A e. RR /\ B e. RR ) -> ( A < B <-> ( A <_ B /\ B =/= A ) ) )
4 1 2 3 mp2an
 |-  ( A < B <-> ( A <_ B /\ B =/= A ) )